Local and non-local vertex corrections in GW for extended and localized systems

A non-local operator like the self-energy can be consistently calculated through many-body perturbation theory for systems of interacting electrons. This is usually done within the framework of Hedin's $GW$ approximation. If the initial Green's function is obtained within a local approximation like DFT-LDA, there is in principle a local vertex given by the static exchange-correlation kernel in the first iteration (Del Sole \emph{et al.} PRB {\bf 49}, 8024 (1994)). We present total energies and bandwidths for jellium and equivalent quantities for He, Be and Ne. We show that a local vertex implemented in both the screened interaction and the self-energy leads to unphysical results. A local vertex in the screened interaction only provides results on par with or slightly better than standard one-shot $G_ {0}W_{0}$. Finally, we obtain significant improvements by introducing non-local vertex corrections derived from non-local starting aproximations for the self-energy.